Optimal. Leaf size=81 \[ \frac {1}{8} x \left (3 a^2+8 a b+8 b^2\right )+\frac {3 a (a+2 b) \sin (e+f x) \cos (e+f x)}{8 f}+\frac {a \sin (e+f x) \cos ^3(e+f x) \left (a+b \tan ^2(e+f x)+b\right )}{4 f} \]
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Rubi [A] time = 0.09, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {4146, 413, 385, 203} \[ \frac {1}{8} x \left (3 a^2+8 a b+8 b^2\right )+\frac {3 a (a+2 b) \sin (e+f x) \cos (e+f x)}{8 f}+\frac {a \sin (e+f x) \cos ^3(e+f x) \left (a+b \tan ^2(e+f x)+b\right )}{4 f} \]
Antiderivative was successfully verified.
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Rule 203
Rule 385
Rule 413
Rule 4146
Rubi steps
\begin {align*} \int \cos ^4(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b+b x^2\right )^2}{\left (1+x^2\right )^3} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {a \cos ^3(e+f x) \sin (e+f x) \left (a+b+b \tan ^2(e+f x)\right )}{4 f}+\frac {\operatorname {Subst}\left (\int \frac {(a+b) (3 a+4 b)+b (a+4 b) x^2}{\left (1+x^2\right )^2} \, dx,x,\tan (e+f x)\right )}{4 f}\\ &=\frac {3 a (a+2 b) \cos (e+f x) \sin (e+f x)}{8 f}+\frac {a \cos ^3(e+f x) \sin (e+f x) \left (a+b+b \tan ^2(e+f x)\right )}{4 f}+\frac {\left (3 a^2+8 a b+8 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (e+f x)\right )}{8 f}\\ &=\frac {1}{8} \left (3 a^2+8 a b+8 b^2\right ) x+\frac {3 a (a+2 b) \cos (e+f x) \sin (e+f x)}{8 f}+\frac {a \cos ^3(e+f x) \sin (e+f x) \left (a+b+b \tan ^2(e+f x)\right )}{4 f}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 58, normalized size = 0.72 \[ \frac {4 \left (3 a^2+8 a b+8 b^2\right ) (e+f x)+a^2 \sin (4 (e+f x))+8 a (a+2 b) \sin (2 (e+f x))}{32 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.39, size = 62, normalized size = 0.77 \[ \frac {{\left (3 \, a^{2} + 8 \, a b + 8 \, b^{2}\right )} f x + {\left (2 \, a^{2} \cos \left (f x + e\right )^{3} + {\left (3 \, a^{2} + 8 \, a b\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{8 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.59, size = 93, normalized size = 1.15 \[ \frac {{\left (3 \, a^{2} + 8 \, a b + 8 \, b^{2}\right )} {\left (f x + e\right )} + \frac {3 \, a^{2} \tan \left (f x + e\right )^{3} + 8 \, a b \tan \left (f x + e\right )^{3} + 5 \, a^{2} \tan \left (f x + e\right ) + 8 \, a b \tan \left (f x + e\right )}{{\left (\tan \left (f x + e\right )^{2} + 1\right )}^{2}}}{8 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.23, size = 78, normalized size = 0.96 \[ \frac {a^{2} \left (\frac {\left (\cos ^{3}\left (f x +e \right )+\frac {3 \cos \left (f x +e \right )}{2}\right ) \sin \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )+2 a b \left (\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+b^{2} \left (f x +e \right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 87, normalized size = 1.07 \[ \frac {{\left (3 \, a^{2} + 8 \, a b + 8 \, b^{2}\right )} {\left (f x + e\right )} + \frac {{\left (3 \, a^{2} + 8 \, a b\right )} \tan \left (f x + e\right )^{3} + {\left (5 \, a^{2} + 8 \, a b\right )} \tan \left (f x + e\right )}{\tan \left (f x + e\right )^{4} + 2 \, \tan \left (f x + e\right )^{2} + 1}}{8 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.57, size = 76, normalized size = 0.94 \[ x\,\left (\frac {3\,a^2}{8}+a\,b+b^2\right )+\frac {\left (\frac {3\,a^2}{8}+b\,a\right )\,{\mathrm {tan}\left (e+f\,x\right )}^3+\left (\frac {5\,a^2}{8}+b\,a\right )\,\mathrm {tan}\left (e+f\,x\right )}{f\,\left ({\mathrm {tan}\left (e+f\,x\right )}^4+2\,{\mathrm {tan}\left (e+f\,x\right )}^2+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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